The thesis is concerned with the introduction of the CG1-DG2 method and the design of an hp-adaptive algorithm in the context of convection-dominated problems in 2D. The CG1-DG2 method combines the continuous Galerkin (CG) method with the discontinuous Galerkin (DG) method by enriching the continuous linear finite element (CG1) space with discontinuous quadratic basis functions. The resulting finite element approximation is continuous at the vertices of the mesh but may be discontinuous across edges. Analysis of the CG1-DG2 discretization in the context of a scalar advection equation shows that the use of upwind-biased convective fluxes leads to an approximation which is stable and exhibits the same convergence rates as the quadratic discontinuous (DG2) method. However, the CG1-DG2 space has fewer degrees of freedom than the DG2 space. In the case of Poisson's equation different strategies known from the DG method can be adopted to approximate the numerical fluxes: the symmetric and non-symmetric interior penalty method as well as the Baumann-Oden method. A priori error estimates for the DG2 method can be shown to hold for the CG1-DG2 approximation as well. Numerical studies confirm that the proposed method is stable and converges at the same rate as the fully discontinuous piecewise-quadratic version. We also present an extension of the CG1-DG2 method to solve the Euler equations and show numerical results which indicate that the CG1-DG2 method gives results similar to those obtained by the DG method. The second part of this thesis presents an hp-adaptive framework for convection-dominated problems. The idea of this algorithm is to divide the mesh in smooth and non-smooth parts, where the smoothness refers to the regularity of the approximated solution. In smooth parts the polynomial degree is increased (p-adaptivity) whereas in non-smooth parts h-adaptivity for linear elements is used. Hereby, a parameter-free regularity estimator is used to determine the smoothness of a function and its gradient by comparing those with reconstructed approximations. In smooth elements the CG1-DG2 method is used. In non- smooth elements a flux-corrected transport scheme is applied and combined with h-adaptivity based on the so-called reference solution approach. Numerical experiments are performed for advection and advection-diffusion equations. Those show the advantage of the hp-adaptive algorithm over pure h-refinement in the context of FCT schemes.